Crack the eggs into a large bowl and add the sugar, oil, orange juice and vanilla extract. Beat for a few minutes until you get a smooth, slightly foamy mixture.
In another bowl, whisk together the flour, baking powder, cinnamon and salt. Gradually fold the dry ingredients into the wet mixture, mixing on low speed just until no lumps remain.
Peel the carrots and grate them on the fine side of the grater, then gently squeeze out any excess juice if there's too much.
Add the grated carrots and chopped walnuts to the batter and fold them in with a spatula, using gentle bottom-to-top motions, until evenly distributed.
Prepare a round cake pan about 8 inches (21 cm) in diameter: line the bottom with parchment paper and lightly grease the sides with a thin layer of oil.
Pour the batter into the pan, smooth the top with a spatula, and bake in a preheated oven at 340°F (170°C). Bake for 45–50 minutes, or until a toothpick inserted in the center comes out clean.
Remove the cake from the oven, let it rest in the pan for 10 minutes, then transfer it to a wire rack and let it cool completely before frosting.
For the frosting, place the cream cheese and softened butter in a bowl. Add the powdered sugar and vanilla extract, then beat on high speed for 5–7 minutes, until you get a smooth, fluffy, glossy frosting.
Once the cake has cooled, trim off the domed top to level it. Then slice it horizontally into 3 roughly equal layers.
Place the first cake layer on a serving plate. Spread part of the frosting over it (about a third, or slightly less) in an even layer.
Stack the second cake layer on top, spread with frosting again, then repeat with the third layer. Use the remaining frosting to cover the top and sides of the cake.
Refrigerate the cake for at least 1–2 hours, to let the frosting firm up and the flavors settle.
Before serving, decorate the cake with chopped walnuts sprinkled on top or around the edges. Slice and serve the cake cold — it's a fragrant, moist dessert with a slightly crunchy texture from the walnuts.